Early undergraduateAnalysisPlanned

Calculus of one variable

Calculus I–II: limits and continuity with ε–δ, derivatives and their applications, the mean value theorem, techniques of integration, improper integrals, series and convergence tests, and Taylor series.

math.analysis.single-variable-calculus

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Builds on

Introductory calculus Exponential and logarithmic functions Sequences and series Trigonometry

Required prerequisites, all the way down: 34

You will learn to

Differentiate and integrate elementary functions using the chain rule, substitution and integration by parts.
State and apply the mean value theorem and L'Hôpital's rule.
Test series for convergence and compute Taylor polynomials with error bounds.

Definition

f is continuous at a if for every ε > 0 there is δ > 0 such that |f(x) − f(a)| < ε whenever |x − a| < δ.
The Taylor series of f at a is Σ f⁽ⁿ⁾(a)(x − a)ⁿ/n!.

Theorem

Mean value theorem: if f is continuous on [a, b] and differentiable on (a, b), then f′(c) = (f(b) − f(a))/(b − a) for some c in (a, b).
Taylor's theorem with remainder.
eˣ = Σ xⁿ/n! for every real x.
An absolutely convergent series converges.

Common misconceptions

Assuming every smooth function equals its Taylor series; f(x) = e^(−1/x²) with f(0) = 0 is a counterexample at 0.
Applying L'Hôpital's rule to limits that are not indeterminate forms.

Leads to

Multivariable and vector calculus Ordinary differential equations Probability theory Numerical methods Real analysis

Optional discoveries

Complex analysis Why does the Taylor series of 1/(1 + x²) at 0 converge only for |x| < 1, although the function is smooth on all of ℝ? Its complex poles at ±i explain it.